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The spatial spread of altruism versus
the evolutionary response of egoists
Jacob C. Koella
Laboratoire d’Ecologie, CC237, Universite
¨
Pierre & Mari e Curie, CNRS UMR 7625, 7 quai St Bernard, 75252 Paris Cedex 5, France
(
jkoella@snv.j ussieu.fr
)
Several recent models have shown that altru ism can spread in viscous populations, i.e. in spatially
structured populations within which individuals interact only with their immediate neighbours and
disperse only over short distances. I ¢rst con¢rm this result with an individual-based model of a viscous
population, where an individual can vary its level of investment into a behaviour that is bene¢cial to its
neighbours but costly to itself. Two distinct classes of individuals emerge: egoists with no or very little
investment into altruism, and altruists with a high level of investment; intermediate le vels of altruism are
not maintained. I then extend the mo del to investigate the consequenc es of letting interaction and
dispersal distances evolve along with altruism. Altruists maintain short distances, while the egoists
respond to the spread of altruism by increasing their interaction and dispersal distances. T his allows the
egoistic individuals to be maintained in the population at a high frequency. Furthermore, the coevolution
of investment into altruism and interaction distance can lead to a stable spatial pattern, where stripes of
altruists (with local interactions) alternate with stripes of egoists (with fa r-reaching interactions). Perhaps
most importantly, this approach shows that the ease with which altruism spreads in viscous populations is
maintained despite countermeasures evolved by egoists.
Keywords:
altruism; cooperation; spatial games; dispersal distance; cellular automata
1. INTRODUCTION
The spread and stability of altruistic behaviour remain
unresolved puzzles in evolutionary ecology. Why should
an individual increase another individual’s ¢tness at the
expense of its own? Indeed, sel¢sh individuals should
always have h igher ¢tness than cooperators, since they
receive the bene¢ts of cooperation without having to pay
the cost. One would therefore predict that any coopera-
tive strategy would soon be eliminated by natural selec-
tion.
Nevertheless, cooperation appears to be widespread
among animals (Dugatkin 1997) and is thought to have
been a driving force in several of the major transitions in
evolution, e.g. the evolution of multicellularity from
unicellular ancestors (Buss 1987; Maynard Smith &
Szathma
¨
ry 1995).
One of the ¢rst prop ositions to solve t his dilemma was
based on kin selection (Hamilton 1963): a gene promoting
altruism may increase its frequency if the bene¢ciary also
happens to carry it. This i s likely to be the case if the
bene¢ciary is a close relative of the altr uist. In many
cases of cooperative behaviour, however, interacting indi-
viduals do not appear to be related. Thus, mechanisms
that do not require kin selection are needed.
One of these mechanisms relies on the assumptions that
individuals repeatedly interact with the same partner and
that each partner’s strategy can take into account the past
history of their interactions. These assumpt ions often lead
to the spread of altruistic strategies (Nowak & Sigmund
1993; Roberts & Sherratt 1998), the best known being tit-
for-tat (Axelrod & Hamilton 1981): cooperate in the ¢rst
interaction and copy your partner’s move for all subse-
quent interactions.
A second mechanism is based on group selection rather
than on repeated interactions between pairs of indivi-
duals. If subpopulations with altruistic individuals fare
better than subpopulations dominated by egoists, the
proportion o f altruists in the population as a who le may
increase despite the altruists having lower ¢tness than
egoists in direct competition between the two (Haldane
19 32; Maynard Smith 1964; Wilson 1977). Recently, the
approach involving discrete subpopulations, where local
population growth alternates with dispersal connectin g
subpopulations, has been modi¢ed to study the evolution
of altruism in viscous populations ( i.e. populations
without discrete subpopulations, but nevertheless with
limited dispersal b ecause o¡spring tend to remain close to
their parents). One of the motivations for this approach is
that it moves the level of selection from the group to t he
individual. In the models so far published, each indivi-
dual interacts with its immediate neighbours and its
reproductive success is evaluated by the number of co-
operators and defectors in its neighbourhood. Though
earlier attempts at models that allow altruism to spread
in viscous populations have failed (Taylor 1992
a
,
b
; Wilson
et al
. 1992), a more recent model has shown elegantly the
possibility of invasion of altrui sm using pai r approxima-
tions of the dynamics (Van Baalen & Rand 1998). These
results are corroborated by appro aches based on spatial
lattices of cellular automata (K illingback
et al
. 1999;
Nowak & May 1992); these di¡er from the viscous popu-
lations in that individuals do not reproduce or disperse
but rather are replaced by individuals with the highest
success within the neighbourhood. In all of these models,
altruistic behaviour ca n spread un der very general
assumptions, including in situations where individuals
from two species show altruistic behaviour, i.e. mutual-
isms (Doebeli & Knowlton 1998).
Thus, the recent models based on spatial systems
suggest that altruism may not be such a di¤cult evolu-
tionary d ilemma after all. However, the models also
Proc. R. Soc. Lond. B (2000) 267, 1979^1985 1979 © 2000 The Royal Society
Received 2 March 2000 Accepted 13 June 2000
doi 10.1098/rspb.2000.1239
suggest that altruists can invade only if their dispersal
distance and the distance over which th ey interact with
neighbours is limited. Thus, it is reasonable to expect that
evolutionary pressure will cause egoists to evolve further
dispersal and interaction distances in response to the
spread of altruism, so that they are more e¤cient at
invading altruistic clusters (Van Baalen & Rand 1998).
But what are the consequences of such an evolutionar y
response by the egoists, and in particular, will it lead to
exclusion of altruism? The two possibilities are that
altruistic strategies are elimin ated from the population by
the evolutio nary response of the egoists or, alternative ly,
that two classes of individuals are formed: altruists with
very local interactions and egoists with far-reaching
dispersal and interactions.
I investigate these questions by modelling a viscous
population of cellular automata. Following several recent
models, I allow investment into altruism to vary as a
continuous trait (Killingback
et al
. 1999; Sherratt &
Roberts 1998; Va n Baalen 1998; Wahl & Nowak 1999).
Additionally, and in contrast to these models, I allow the
simultaneous evolution of the dispersal distance of
o¡spring and the distance over which individuals interact
with their neighbours. In doing so, I have two goals: ¢rst,
to con¢rm the results reached with a pair approximation
of the viscous population (Van Baalen & Rand 1998)
and, second, to investigate the destabilizing consequences
of the simultaneous evolution of investment into altruism,
dispersal distance and interaction distance.
2. GENERAL MODEL
In all of the models presented here, individuals interact
with one another on a spatial lattice (¢gure 1). The lattice
has the shape of a torus, i.e. individuals on the edges
interact with individuals on opposite edges, eliminating
edge e¡ects. Every site on the lattice can be either empty
or occupied by an individual investing level
I
into
altruistic behaviour. Investment into altruism takes values
between 0 and 0.99, with random mutations changing
the investment in steps of
§
0.01. Individuals interact
with their neighbours up to a distance that is called the
interaction distance.
The state of the lattice changes at each time-step as a
consequence of random birth and death eve nts. The prob-
ability of death is constant and identical for each indivi-
dual (i.e. independent of the chosen strateg y). An
individual’s probability of reproducing, however, is deter-
mined by its own strategy and by its neighbourhood.
Speci¢cally, investment into altruism incurs a cost,
c
,
which is assumed to be proportional to the investment
(i.e.
c
ˆ
®
I
), and reproduction increases with the average
level of the neighbours’ investment into altruism
according to the function (1
¡
e
¡
k
·
I
). Thus, the birth rate
of an individual can be de¢ned as
b
(
I
)
ˆ
b
0
‡
(1
¡
e
¡
k
·
I
)
¡
®
I
, (1)
where
b
0
is the probability of reproduction without any
interactions and the bene¢t due to the
N
neighbours’
investment is calculated as the mean level of investment
of individuals within the focal individual’s interaction
distance, i.e.
·
I
ˆ
I
i
/
N
. Alternative bene¢t functions,
e.g. a fu nction where an individual’s reproduction
increased proportionally to its neighbours’ investment,
gave similar results.
The o¡spring are successful only if they ¢nd an empty
site within dispersal distance of their parent. Each empty
site is occupied by the o¡spring of the parent with l ocally
(i.e. within dispersal di stance) the highest probability o f
reproduction. (Alternatively, o¡spring could be chosen
randomly with probability proportional to the birth rate
of the parents within dispersal distance of the site; this
gave similar results, though evolution was slower due to
increased drift.)
Finally, mutations modify the genetic basis of the indi-
viduals’ behaviour. They a re assumed to occur randomly
with a low probability (in all simulations shown, the
probability was set to 0.001) and to have a small e¡ect,
i.e. they can change the investment into altruism by only
one step in either direction.
(a) Results of the general model
In all of the simulations, the si ze of the lattice is set to
50
£
50 individuals and the individuals are initialized
with no investment into altruism.
The typical results of simulations con¢rm previously
established results: investment into altruism spreads
readily in a population when dispersal distance is low
(¢gure 2
a
). The system eventually reaches a dynamic
equilibrium, where the mean level of investment in the
population remains more or less constant (¢gure 2
a
). At
this equilibriu m, the population is divided into a group of
individuals with almost no level of investment into
altruism (egoists) and a distinct group wit h considerable
investment into altruism (altruists); no individuals have
intermediate levels of investment (¢ gure 2
b
). As dispersal
1980 J. C. Koella
Spatial spread of altruism versus evolutionary response of egoists
Proc. R. Soc. Lond. B (2000)
Figure 1. Example of a spatial lattice. O ccupied spaces are
shown by circles. A focal individual is shown in black. It
interacts with its neighbours (shown in grey) up to the
interaction distance of 2; its o¡spring can potentially disperse
to the empty sites (marked by crosses) up to the dispersal
distance of 3. Note that the local environment of a focal
individual (f or in teraction and dispersal) is a ssumed to form a
square.
distance increases altruism can spread if the bene¢t gained
by altruistic neighbours is su¤ciently high (¢gure 2
d
). The
mean level of investment into altruism within the popula-
tion reaches its maximum at an intermediate va lue of
`initial bene¢t’ (¢gure 2
d
). Note that the term `initial
bene¢t’ in ¢gure 2
d
refers to the slope of the bene¢t func-
tion when all neighbours are egoistic, i.e.
k
in equation
(1). Therefore, as initial bene¢t (the slope of th e function)
in equation (1) increases, a birth rate close to the maximal
value is reached with lower levels of altruism.
Furthermore, the altruists form clusters of individuals
that pro¢t from their altruistic behaviour while remaining
resistant to invasion by the egoists they are surrounded by
(¢gure 2
c
), again con¢rming previous results (Killing-
back
et al
. 1999; Van Baalen & Rand 1998). The mechan-
isms leading to the spread of altruistic clusters have been
described ea rlier (Killingback
et al
. 1999; Van Baalen &
Rand 1998) and will not be discussed here; rather, I will
in ½½ 3 and 4 concentrate on the e¡ect of coevolution with
dispersal and interaction di stance, respectively.
3. COEVOLUTION OF DISPERSAL DISTANCE
In a ¢rst mo di¢cation of the basic model described
above, the maximal dispersal distance of o¡spring is
allowed to evolve with rando m mutations changing
dispersal distance by
§
1 (while the interaction distance
is maintained at a constant value of one spatial unit). As
dispersal in models with no selection pressure is likely to
drift into the population (and thus to prevent the spread
of altruistic behaviour), a small cost of dispersal is
subtracted from the equation de¢ning the number of
o¡spring. Thus, the birth function becomes
b
(
I
)
ˆ
b
0
‡
(1
¡
e
¡
k
·
I
)
¡
®
I ¡
¯
d
, (2)
where
d
is the dispersal distance and ¯ is the cost of
dispersal (which is assumed to be 0.01 in all si mulations).
As in the previous model, the population is initialized
with egoistic individuals that invest nothing into altruism.
Spatial spread ofaltruism versus evolutionary response of egoists
J. C. Koella 1981
Proc. R. Soc. Lond. B (2000)
0
20
40
60
25
50
0 10 000 20 000 30 000
mean investment (
´
100)
time
0 10 20 30 40 50 60 70
100
200
300
400
number of individuals
investment into altruism (
´
100)
(
a
)
(
c
)
(
b
)
(
d
)
0
10
20
30
1 10 100 500
investment (
´
100)
initial benefit of altruism
2
3
1
Figure 2. Typical results of simulations using the basic model where only investment into altruistic behaviour is allowed to
evolve. (a) Evolution of the average investment into altruism. (b) Distribution of investment reached after 30 000 generations.
(c) Spatial distribution of investment reached after 30 000 generations. The square represents the 50
£
50 grid. As indicated in
the legend, dark patches represent egoistic individuals and light patches represent altruistic individuals. (d ) Average investment
into altruism reached after 100 000 generations for various slopes of the bene¢t function (product of k and in equation (1))
and dispersal distances 1 (dotted line), 2 (dashed line) and 3 (solid line). As investment £uctuates because of the randomness
of the processes, the mean of the last 50 000 generations is shown. The parameters in the simulations are size of arena, 50
£
50;
fecundity in the absence of any interactions, b, 1; maximal bene¢t of altruism, , 5; slope of cost of altruism, ®, 1; death rate, 0.1;
interaction distance, 1; and mutation rate, 0.001. In (a
^
c) the slope of the bene¢t of altruism at zero investment (k ) is 5 and the
dispersal distance is 1. Note that in (d ) the parameter is maintained at 5 while k is allowed to vary.
Additionally, the dispersal distance is set to 1 (the lowest
possible value) in the initial generation.
When the bene¢t of altruism is su¤ciently high, invest-
ment into altruism can again increase (¢gure 3
a
).
However, the egoistic individuals (ma intaining low levels
of investment into altruism) respond to the spread of
altruism by increasing their dispersal dista nce. This
allows them to invade altruistic clusters more e¡ectively
and keep s the altruistic individuals from evolving high
levels of investment. As altruistic individua ls disperse
only short distances to maintain the integrity of their
local clusters, the evolutionary response of the egoists to
increasing levels of altruism generates a negative correla-
tion between dispersal di stance and investment into
altruism (¢gure 3
b
). Thus, the discrete distributions of
investment formed for egoistic and altruistic individuals
(¢gure 3
c
) are re£ected in their dispersal d istances. At the
end of the simulations shown in ¢g ure 3, about 41% of
the altruistic individuals spread their o¡spring by only
one spatial unit, while 97% of the egoistic individuals
had a dispersal distance greater than one unit. Addition-
ally, none of the altruists spread their o¡spring more than
two units, while almost a quarter of the egoistic indivi-
duals spread their o¡spring over three or four units. The
increased dispersal changes the spatial pattern: while the
clustered pattern of altruists and egoists remains
apparent, the cluste rs tend to become larger than for the
basic model (¢ gure 3
d
).
4. COEVOLUTION OF INTERACTION DISTANCE
In a second modi¢cation of the basic model, the
dispersal distance i s held constant (one unit in all simula-
tions) but instead the interaction distance of individuals is
allowed to evolve with random mutations changing it by
§
1 (see ¢gure 1). There is no direct cost o f interacting, so
that the birth function (equation (1)) is used.
The population is initialized with egoistic individuals
that interact only with their immediate neighbours. As in
the basic model, the mean level of investment increases
rapidly to an equilibriu m level that is maintained by the
dynamics of the spatial system (¢gure 4
a
) and two
discrete groups of individuals are formed with either very
low or quite high levels of investment into altruism
1982 J. C. Koella
Spatial spread of altruism versus evolutionary response of egoists
Proc. R. Soc. Lond. B (2000)
0 5 10 15 20
0
100
200
300
400
500
600
700
number of individuals
investment into altruism (
´
100)
5
10
1
2
0 10 000 20 000 30 000
investment into altruism (
´
100)
dispersal distance
time
altruism
dispersal
0
5
10
15
1
-
-
-
-
-
0.8
0.6
0.4
0.2
0
0 10 000 20 000 30 000
correlation
time
(
a
) (
b
)
(
c
)
(
d
)
Figure 3. Typical results of simulations using the model where dispersal distance can evolve as well as investment into
altruistic behaviour. (a) Evolution of the average investment into altruism (solid line) and average dispersal distance (dotted
line). (b) Evolution of the correlation between dispersal distance and investment into altruism. (c) Distribution of investment
reached after 30 000 generations. (d ) Spatial distribution of investment reached after 30 000 generations. The square represents
the 50
£
50 grid. As indicated in the legend, dark patches r epresent egoistic individuals and light patches represent altruistic
individuals. The parameters in the simulations are as in ¢gure 2, except that dispersal distance is not held constant and the cost
of dispersal (¯) is 0.01.
(¢gure 4
c
). A strong negative correlation between invest-
ment into altruism and interaction distance shows that
while altruistic individuals evolve to interact only with
their immediate neighbours, the egoists evolve a great ly
increased interaction distance (¢gure 4
b
). Finally, and in
contrast to the previous models, the coevolution of inter-
action dista nce and investment into altruism leads to a
stable spatial pattern with alternating stripes of egoists
and altruists (¢gure 4
d
). The direction of the stripes
(horizontal or vertical) depends on the initial conditions.
Simulations where the arena does not have periodic
boundary conditions (i.e. where individuals on an edge
cannot interact with or disperse to the opposite edges)
reveal the same pattern (results not shown).
5. DISCUSSION
The main results of the described models are: ¢rst, that
investment into a ltruism can evolve to a high level under
rather general co nditions; second, that egoists cannot be
eliminated from the population, but rather respond to the
spread of altruism by increasing their dispersal distance
and the distance over which they interact with their
neighbours; and third, that the combined evolution o f
investment into a ltruism and the interaction distance ca n
lead to stable spatial patterns.
When investment into altr uism can evolve, but dispersal
and interaction distances remain constant, the results of
the model con¢rm previous attempts at describing the
spread of altruism in spatially structured populations
(Killingback
et al
. 1999; Nowak & May 1992; Van Baalen
& Rand 1998). Altruism can spread if the b ene¢t to the
recipient of the altruistic behaviour is su¤ciently high to
compensate for the cost of behaving altruistically. As
described in detail p reviously (Killingback
et al
. 1999; Van
Baalen & Rand 1998), the success of altruistic behaviour
can be explained with group-selection arguments. In
viscous populations such as the ones described in this
paper, individuals sharing an a ncestor (and thus with
similar traits) form clusters because of limited dispersal.
As cooperation within groups of individuals interact ing
only among themselves bene¢ts all individuals in the
group, a cluster of altruistic individual s has a higher rate of
spread than a cluster of egoistic individuals, which leads to
Spatial spread of altruism versus evolutionary response ofegoists
J. C. Koella 1983
Proc. R. Soc. Lond. B (2000)
0
10
20
30
1
2
3
4
0 10 000 20 000 30 000
investment into altruism (
´
100)
interaction distance
time
altruism
interaction
0
20
40
60
0 10 000 20 000 30 000
time
0 20 40 60
0
200
400
600
800
1000
number of individuals
investment into altruism (
´
100)
(
c
)
1
-
-
-
-
-
0.8
0.6
0.4
0.2
0
correlation
(
a
) (
b
)
(
d
)
Figure 4. Typical results of simulations using the model where interaction distance can evolve a s well as investment into
altruistic behaviour. (a) Evolution of the average investment into altruism (solid line) and average interaction distance (dotted
line). (b) Evolution of the correlation between interaction distance and investment into altruism. (c) Distribution of investment
reached after 30 000 generations. (d ) Spatial distribution of investment reached after 30 000 generations. The square represents
the 50
£
50 grid. As indicated in the legend, dark patches represent egoistic individuals and light patches represent altruistic
individuals. The parameters in the simulations are as in ¢gure 2, except that interaction distance is not held constant.
the spread of altruism in the population as a whole.
However, altruism can spread only when interactions are
local, i.e. when the distance over which individuals
interact with their neighbour s and the distance their
o¡spring disperse are small. Furthermore, egoistic cheaters
can generally invade clusters of altruists because of the
reduced cost of their behaviour, so that the population as a
whole is held in a dynamic equilibrium at an intermediate
investment into altruism. What has received less attention
in earlier literature is that the level of investment into
altruism is bimodal: while some individuals invest very
little into altruism, others invest very high levels; inter-
mediate levels of investment are not ma intained.
When, in addition to investment into altruism,
dispersal distance or distance over which individuals
interact with their neighbours can respond to natural
selection rather tha n being ¢xed, egoistic individuals can
respond to the spread of altruists and thus limit the level
and prevalence of altruism.
One of the responses of egoists to the spread of altruism
is to increase their dispersal distance. This enables them
to penetrate the interio rs of altruistic clusters and thereby
to increase their likeliho od of reproducing and spreading.
At the population level, the increased dispersal distance
increases the mixing of the population, which makes it
more di¤cult for altruistic clusters to form and thus for
altruism to spread. Furthermore, the simultaneous evolu-
tion of dispersal distance and investment into altruism
leads to the coexistence of two types of individuals,
rapidly moving egoists and sedentary altruists, as
predicted in earlier studies (Van Baalen & Rand 1998).
However, the separation b etween altruists and egoists is
not as clear as when altruism alone evolves, as the
increased dispersal distance of the egoistic individuals
maintains a low level of investment by the altruists.
Egoists can also respond to the spread of a ltruism by
increasing the distance over which they interact with
their neighbours, as egoists with far-reaching interactions
may bene¢t from many altruistic interactions while not
having to pay the cost of altruism. In contrast, altruists
maintain close interactions within their local clusters, so
that they run less risk of interacting with egoists outside
their cluster. This result is reminiscent of the simultaneous
evolution of altruism with choosiness (Sherratt & Roberts
1998): if individuals can choose with whom to interact,
altruism is more l ikely to evolve. In the model described
in this paper, altruists are choosy in the sense that they
interact only with their i mmediate neighbours, while
egoists i nteract with a larger (a nd thus more diverse)
sample of neighbours. This choosiness leads to two clearly
distinct classes of individuals: egoistic individuals with
far-reaching interactions and altruists with local inter-
actions. This, in turn, leads to a stable spatial pattern of
altruism. Indeed, the interactions between egoistic and
altruistic individuals are similar to the frequency-
dependent interactions between two discrete species,
which lead to similar spatial patterns (Molofsky
et al
.
1999). More generally, t he patterns found here and in
earlier studies (Molofsk y
et al
. 1999), which describe inter-
actions that are governed by random processes, re£ect the
spatial patterns that can be found in deterministic systems
of interacting populations (Hassell
et al
. 1994; Molofsky
1994; Okubo 1980).
Overall, the model presented here, together with
previous models of the spread of co operation in spatially
structured populations (Killingback
et al
. 1999; Nowak &
May 1992; Van Baalen & Rand 1998), provides a basis for
the understanding of cooperative and altruistic b ehaviour.
Together, these models show that high levels of investment
in altruistic behaviour can evolve with relative ease from
a population of egoists, and that the evolutionary pro cess
will generally lead to a mixture of highly altruistic and
highly egoistic individuals with very few intermediate
behaviours.
Naturally (though this has not prev iously been consid-
ered), the egoists will evolve additio nal strategies to
exploit altruists and thus to check their spread; in the
model presented here they respond by increasing their
dispersal or interaction distance. Despite this evolu-
tionary response, altruistic behaviour is ma intained in
the population; in fact, though either evolutionary
response decreases the prevalence of altruism, evolution
of interaction distance tends to enhance the di¡erence in
investment levels between altruists and egoists and to
stabilize the spatial pattern.
Thus, modelling the coevolution of investment into
altruistic behaviour and the response by the egoists may
help to understand more thoroughly not only the invasion
but also the stability o f cooperative behaviour observed at
all levels of biological organization, from the cooperation
between replicating molecules (Eigen & Schuster 1979;
Maynard Smith & Szathma¨ ry 1995) to the social organi-
zation of organisms (Dugatkin 1997).
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J. C. Koella 1985
Proc. R. Soc. Lond. B (2000)
